Related papers: On the asymptotics of real solutions for the Painl…

In this paper, we revisit large variable asymptotic expansions of tronqu\'ee solutions of the Painlev\'e I equation, obtained via the Riemann-Hilbert approach and the method of steepest descent. The explicit construction of an extra local…

In this paper, we study the asymptotic behavior and connection problem of Painlev\'e I (PI) equation through a detailed analysis of the Stokes multipliers associated with its solutions. Focusing on the regime where the derivative at the…

The present paper gives an overview of the recent developments in the description of critical behavior for Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations. It was conjectured that this behavior…

The paper concerns asymptotic studies for the sixth Painlev\'e transcendent as independent variable tends to infinity. The primary tool is averaging and the Whitham method. Elliptic ansatz, appropriate modulation equation and asymptotics…

We consider a connection problem of the first Painlev\'{e} equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real…

In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{\'e} equations. Such problems can be characterized by the question as to how the…

We consider a family of solutions to the Painlev\'e II equation $$ u''(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou nonlinear steepest…

The 0-instanton solution of Painlev\'e I is a sequence $(u_{n,0})$ of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and 2-dimensional quantum gravity. The…

We consider three special cases of the initial value problem of the first Painlev\'e equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of…

In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlev\'e transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the…

The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of DP1 are classified under criterion of their behavior while argument tends to infinity. The Isomonodromic Deformations Method yields asymptotic…

We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the…

For the first Painlev\'e transcendents Kitaev established an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips near the point at infinity. We present an explicit error bound of this asymptotic…

Unstable separatrix solutions for the first and second Painlev\'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise…

We study the asymptotic behaviour of solutions of the fourth Pain\-lev\'e equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show…

For the fourth Painlev\'e transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.

We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlev\'e equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each…

This paper proposes a new approach to the asymptotic analysis of Painlev\'e equations. The approach is based on representing solutions of the Painlev\'e equations using formal series in two variables, $\sum_{k=0}^{\infty}y^kA_k(x)$, with…

This paper is a continuation of our analysis, begun in arXiv:1310.2276, of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of…

In this study, we consider the asymptotic behaviour of the first discrete Painlev\{e} equation in the limit as the independent variable becomes large. Using an asymptotic series expansion, we identify two types of solutions which are…